Question

What type of conic section is given by the equation y^2-9x^2=900 and what is the domain and range? Show your work.

Answer 1

Answer:

Domain : x ∈ Real Numbers

Range : y ≥ 30

Step-by-step explanation:

The given equation is:

$y^2-9x^2=900$

Simplifying it

$\frac{y^2}{900}-\frac{9x^2}{900}=\frac{900}{900}\\\frac{y^2}{900}-\frac{x^2}{100}=1\\\frac{y^2}{30^2}-\frac{x^2}{10^2}=1$

Where Standard equation of parabola is:

$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

Which are similar. Conic Section is a parabola.

Find Domain and Range:

Simplify the given equation:

$y^2-9x^2=900\\y=\sqrt{9x^2+900},y=-\sqrt{9x^2+900}\\y=3\sqrt{x^2+100},y=-3\sqrt{x^2+100}$

For whatever value of x, term under the square root always remains positive, so

Domain : x ∈ Real Numbers

For minimum value of x i.e 0, y=30. If we increase x, y also increases. So

Range : y ≥ 30